关于复射影空间中极小子流形的某些整体Pinching定理

On some global Pinching theorems for minimal submanifolds of a complex projective space

本文利用对复射影空间中紧致极小子流形的第二基本形式长度平方进行积分形式的估计方法 ,证明了复射影空间中紧致复子流形和紧致全实极小子流形的整体

Let \$M\+n\$ be a compact minimal submanifold of a complex projective space \$CP\+\{n+p\}\$, and \$σ\$ the second fundamental form of \$M\+n\$. In this note, the results of \ were improved and the following global Pinching theorems were obtained.\;Theorem 1: Let \$M\+n\$ be a complex \$n\$\|dimentional compact complex submanifold in \$CP\+\{n+p\}(n≥2).\$ Let \$σ\$ the second fundamental form of \$M\$. Then there is a comtant \$A(n)\$ depending only on \$n\$, such that if \$‖|σ|\+2‖\-\{n2\}<A(n)\$, then \$σ≡ 0,i.e ., M\+n\$ must be totally geodesic.\;Theorem 2: Let \$M\+n\$ be an \$n\$\|dimentional compact totally real minimal submanifold in \$CP\+\{n+p\}(n≥3).\$ Let \$σ\$ the second fundamental form of \$M\$. Then there is a constant \$A′(n)\$ depending only on \$n\$, such that if \$‖|σ|\+2‖\-\{n2\}<A′(n),\$ then \$σ≡0, i.e ., M\+n\$ must be totally geodesic. Here\$\$‖f‖\-k=(∫\-Mf\+k)\+\{1/k\}.\$\$